几类真空时空中波动方程的辐射场理论

The stability of Minkowski space-time, Shwarzschild space-time and Kerr space-time are essential problems in general relativity. As a first step of studying the stability problems, the global asymptotic behavior of solutions to the Cauchy problem of linear and some nonlinear wave equations on those three vacuum space-times will be investigated in this program. Radiation field will be used to describe the asymptotic behavior for wave solutions. For the linear case, the coefficient of leading term in the asymptotic expansions for the solution along the light rays will define the radiation field. For the nonlinear case, if the solution globally exists and has asymptotic expansions along the light rays, then the term which have the same order as the leading term in the corresponding linear case will give the definition of the radiation field. Each vacuum space-time will be compactified such that the wave solutions will have simple behavior near the boundary of the compactification. Both forward and backward energy estimates will be obtained for a conformal transformation of the equation on the compactified space-time. From this the mapping property for the map from the Cauchy data to the radiation field will be proved. And the characteristic initial data problems will be studied so that the wave equations can be solved backward.

Minkowski时空，Shwarzschild时空和Kerr时空的稳定性问题是广义相对论中的重要问题。作为研究稳定性问题的第一步，在此项目中将主要研究这三类真空时空的线性波动方程和一些非线性波动方程的Cauchy问题的解的整体渐进行为。辐射场将会被用来描述这些波动方程解的渐进行为。对线性情况，方程的解沿着光线传播方向渐进展开的主项系数即定义为辐射场。对非线性情况，如果解整体存在并沿着光线传播方向有渐进展开，那么和线性情况主项同阶项的系数将会被定义为辐射场。对每一种时空背景下的波动方程，整个时空将会被合理紧化使得解在这些紧化空间上边界处的行为有简单的表达。同时方程本身在进行合理的共形变换后，会在紧化空间上得到向前和向后的能量估计。由此可以给出从Cauchy初值到辐射场的映射的性质，并回答波动方程从辐射场定义处反向解回的问题。

Minkowski时空，Schwarzschild时空和Kerr时空的稳定性问题是广义相对论中的重要问题。作为研究稳定性问题的第一步，在此项目中将主要研究这三类真空时空的线性波动方程和一些非线性波动方程的Cauchy问题的解的整体渐进行为。辐射场将会被用来描述这些波动方程解的渐进行为。我们主要对Minkowski空间上的满足零条件的非线性波动方程和一些半线性波动方程回答了这些问题，同时对Schwarzschild上半线性波动方程部分回答了这些问题。